Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion

Youshan Tao; Michael Winkler

doi:10.3934/dcds.2012.32.1901

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2012, Volume 32, Issue 5: 1901-1914. Doi: 10.3934/dcds.2012.32.1901

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Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion

Youshan Tao^1, and
Michael Winkler^2,

1.
Department of Applied Mathematics, Dong Hua University, Shanghai 200051
2.
Institut für Mathematik, Universität Paderborn, 33098 Paderborn

Received: October 31, 2010

Revised: September 30, 2011

Published: April 2012

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Abstract

This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled chemotaxis-fluid model $$ \left\{ \begin{array}{l} n_t+ u\cdot \nabla n=\Delta n^m - \nabla \cdot (n\chi(c)\nabla c)\\ c_t+ u\cdot \nabla c=\Delta c-nf(c)\\ u_t +\nabla P-\eta \Delta u+n \nabla \phi=0 \\ \nabla \cdot u=0, \end{array} \right. $$ which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid. The given functions $\chi$ and $f$ are supposed to be sufficiently smooth and such that $f(0)=0$.
It is proved that global bounded weak solutions exist whenever $m>1$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0 \ge 0$ and $c_0\ge 0$. This extends a recent result by Di Francesco, Lorz and Markowich (Discrete Cont. Dyn. Syst. A 28 (2010)) which asserts global existence of weak solutions under the constraint $m \in (\frac{3}{2},2]$.

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Mathematics Subject Classification: Primary: 35K55, 35Q30; Secondary: 35Q35, 92C17.

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